Form Problems after Magento Update to CE 1.8.1 or EE 1.13.1 (Login, Cart)

If the login forms in the frontend do not work anymore after an update to Magento CE 1.8.1 or EE 1.13.1, the used theme has to be updated. These forms now need a form key, if it is missing, validation in the controller fails. The same applies for the “Update Cart” button in cart.phtml. How to fix it, see this link: